Question

If \(25^{3a+5} + 125^{2a+3} = 18750\), what is the value of a?

• A. \( \frac{2}{3} \)
• B. \( -\frac{2}{3} \)
• C. \( \frac{1}{3} \)
• D. \( -\frac{1}{3} \)

Hint:

When you see an equation with different numerical bases that are powers of the same number (like 2, 4, 8, 16 or 3, 9, 27), the first step is always to convert everything to that common base. Then look for opportunities to factor.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This is an equation involving exponents. The strategy is to express all terms with a common base and then use the properties of exponents to solve for the variable.
Step 2: Key Formula or Approach:
Properties of exponents:

\( (x^m)^n = x^{mn} \)
\( x^m \cdot x^n = x^{m+n} \)
The common base for 25 and 125 is 5.
Step 3: Detailed Explanation:
First, rewrite the bases in terms of 5: \[ 25 = 5^2 \] \[ 125 = 5^3 \] Substitute these into the original equation: \[ (5^2)^{3a+5} + (5^3)^{2a+3} = 18750 \] Apply the power of a power rule \( (x^m)^n = x^{mn} \): \[ 5^{2(3a+5)} + 5^{3(2a+3)} = 18750 \] \[ 5^{6a+10} + 5^{6a+9} = 18750 \] The exponents are very similar. We can factor out the term with the smaller exponent, which is \(5^{6a+9}\). \[ 5^{6a+9} (5^1 + 1) = 18750 \] \[ 5^{6a+9} (5 + 1) = 18750 \] \[ 5^{6a+9} (6) = 18750 \] Now, isolate the exponential term by dividing by 6: \[ 5^{6a+9} = \frac{18750}{6} \] \[ 5^{6a+9} = 3125 \] To solve for \(a\), we need to express 3125 as a power of 5. \[ 5^1 = 5 \] \[ 5^2 = 25 \] \[ 5^3 = 125 \] \[ 5^4 = 625 \] \[ 5^5 = 3125 \] So, the equation becomes: \[ 5^{6a+9} = 5^5 \] Since the bases are equal, we can equate the exponents: \[ 6a + 9 = 5 \] \[ 6a = 5 - 9 \] \[ 6a = -4 \] \[ a = -\frac{4}{6} = -\frac{2}{3} \] Step 4: Final Answer:
The value of \(a\) is \(-\frac{2}{3}\).